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Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

Authors Hung Le, Lazar Milenković, Shay Solomon



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Hung Le
  • University of Massachusetts, Amherst, MA, USA
Lazar Milenković
  • Tel Aviv University, Israel
Shay Solomon
  • Tel Aviv University, Israel

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Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 54:1-54:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.54

Abstract

In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(n α_k(n)) edges, for any k≥2. The function α_k is the inverse of a certain Ackermann-style function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α₀(n)=⌈n/2⌉, α₁(n)=⌈√n⌉, α₂(n)=⌈log n⌉, α₃(n)=⌈log log n⌉, α₄(n)=log^* n, α₅(n)=⌊1/2 log^*n⌋, .... Roughly speaking, for k≥2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Also, α_{2α(n)+4}(n)≤4, where α(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases k=2 and k=3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ε>0, any (1+ε)-spanner for the uniform line metric with hop-diameter at most k must have at least Ω(n α_k(n)) edges.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Euclidean spanners
  • hop-diameter
  • inverse-Ackermann
  • lower bounds
  • sparse spanners

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References

  1. Ittai Abraham and Dahlia Malkhi. Compact routing on euclidian metrics. In PODC, pages 141-149. ACM, 2004. Google Scholar
  2. Pankaj K. Agarwal, Yusu Wang, and Peng Yin. Lower bound for sparse euclidean spanners. In SODA, pages 670-671. SIAM, 2005. Google Scholar
  3. Noga Alon and Baruch Schieber. Optimal preprocessing for answering on-line product queries. Technical report, Tel Aviv University, 1987. Google Scholar
  4. Ingo Althöfer, Gautam Das, David P. Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discret. Comput. Geom., 9:81-100, 1993. Google Scholar
  5. Sunil Arya, Gautam Das, David M. Mount, Jeffrey S. Salowe, and Michiel H. M. Smid. Euclidean spanners: short, thin, and lanky. In STOC, pages 489-498. ACM, 1995. Google Scholar
  6. Sunil Arya, David M. Mount, and Michiel H. M. Smid. Randomized and deterministic algorithms for geometric spanners of small diameter. In FOCS, pages 703-712. IEEE Computer Society, 1994. Google Scholar
  7. Yair Bartal, Nova Fandina, and Ofer Neiman. Covering metric spaces by few trees. In ICALP, volume 132 of LIPIcs, pages 20:1-20:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  8. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners. In SODA, pages 932-941. SIAM, 2009. Google Scholar
  9. Hans L. Bodlaender, Gerard Tel, and Nicola Santoro. Trade-offs in non-reversing diameter. Nord. J. Comput., 1(1):111-134, 1994. Google Scholar
  10. Paul B. Callahan and S. Rao Kosaraju. Faster algorithms for some geometric graph problems in higher dimensions. In SODA, pages 291-300. ACM/SIAM, 1993. Google Scholar
  11. T.-H. Hubert Chan and Anupam Gupta. Small hop-diameter sparse spanners for doubling metrics. Discret. Comput. Geom., 41(1):28-44, 2009. Google Scholar
  12. Bernard Chazelle. Computing on a free tree via complexity-preserving mappings. Algorithmica, 2:337-361, 1987. Google Scholar
  13. Bernard Chazelle and Burton Rosenberg. The complexity of computing partial sums off-line. Int. J. Comput. Geom. Appl., 1(1):33-45, 1991. Google Scholar
  14. Danny Z. Chen, Gautam Das, and Michiel H. M. Smid. Lower bounds for computing geometric spanners and approximate shortest paths. Discret. Appl. Math., 110(2-3):151-167, 2001. Google Scholar
  15. Paul Chew. There is a planar graph almost as good as the complete graph. In SCG, pages 169-177. ACM, 1986. Google Scholar
  16. Gautam Das and Giri Narasimhan. A fast algorithm for constructing sparse euclidean spanners. Int. J. Comput. Geom. Appl., 7(4):297-315, 1997. Google Scholar
  17. Gautam Das, Giri Narasimhan, and Jeffrey S. Salowe. A new way to weigh malnourished euclidean graphs. In SODA, pages 215-222. ACM/SIAM, 1995. Google Scholar
  18. Yefim Dinitz, Michael Elkin, and Shay Solomon. Low-light trees, and tight lower bounds for euclidean spanners. Discret. Comput. Geom., 43(4):736-783, 2010. Google Scholar
  19. Michael Elkin and Shay Solomon. Optimal euclidean spanners: Really short, thin, and lanky. J. ACM, 62(5):35:1-35:45, 2015. Google Scholar
  20. Joachim Gudmundsson, Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Approximate distance oracles for geometric graphs. In SODA, pages 828-837. ACM/SIAM, 2002. Google Scholar
  21. Joachim Gudmundsson, Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Approximate distance oracles for geometric spanners. ACM Trans. Algorithms, 4(1):10:1-10:34, 2008. Google Scholar
  22. Joachim Gudmundsson, Giri Narasimhan, and Michiel H. M. Smid. Fast pruning of geometric spanners. In STACS, volume 3404 of Lecture Notes in Computer Science, pages 508-520. Springer, 2005. Google Scholar
  23. Yehuda Hassin and David Peleg. Sparse communication networks and efficient routing in the plane. Distributed Comput., 14(4):205-215, 2001. Google Scholar
  24. Omri Kahalon, Hung Le, Lazar Milenkovic, and Shay Solomon. Can't see the forest for the trees: Navigating metric spaces by bounded hop-diameter spanners. CoRR, abs/2107.14221, 2021. URL: http://arxiv.org/abs/2107.14221.
  25. J. Mark Keil and Carl A. Gutwin. Classes of graphs which approximate the complete euclidean graph. Discret. Comput. Geom., 7:13-28, 1992. Google Scholar
  26. Hung Le, Lazar Milenkovic, and Shay Solomon. Sparse euclidean spanners with tiny diameter: A tight lower bound. CoRR, abs/2112.09124, 2021. URL: http://arxiv.org/abs/2112.09124.
  27. Hung Le and Shay Solomon. Truly optimal euclidean spanners. In FOCS, pages 1078-1100. IEEE Computer Society, 2019. Google Scholar
  28. Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Efficient algorithms for constructing fault-tolerant geometric spanners. In STOC, pages 186-195. ACM, 1998. Google Scholar
  29. Yishay Mansour and David Peleg. An approximation algorithm for minimum-cost network design. In Robust Communication Networks: Interconnection and Survivability, volume 53 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 97-106. DIMACS/AMS, 1998. Google Scholar
  30. Giri Narasimhan and Michiel H. M. Smid. Geometric spanner networks. Cambridge University Press, 2007. Google Scholar
  31. Mihai Patrascu and Erik D. Demaine. Tight bounds for the partial-sums problem. In SODA, pages 20-29. SIAM, 2004. Google Scholar
  32. Satish Rao and Warren D. Smith. Approximating geometrical graphs via "spanners" and "banyans". In STOC, pages 540-550. ACM, 1998. Google Scholar
  33. Shay Solomon. Sparse euclidean spanners with tiny diameter. ACM Trans. Algorithms, 9(3):28:1-28:33, 2013. Google Scholar
  34. Shay Solomon. From hierarchical partitions to hierarchical covers: optimal fault-tolerant spanners for doubling metrics. In STOC, pages 363-372. ACM, 2014. Google Scholar
  35. Shay Solomon and Michael Elkin. Balancing degree, diameter and weight in euclidean spanners. In ESA (1), volume 6346 of Lecture Notes in Computer Science, pages 48-59. Springer, 2010. Google Scholar
  36. Robert Endre Tarjan. Applications of path compression on balanced trees. J. ACM, 26(4):690-715, 1979. Google Scholar
  37. Mikkel Thorup. On shortcutting digraphs. In WG, volume 657 of Lecture Notes in Computer Science, pages 205-211. Springer, 1992. Google Scholar
  38. Mikkel Thorup. Shortcutting planar digraphs. Comb. Probab. Comput., 4:287-315, 1995. Google Scholar
  39. Andrew Chi-Chih Yao. Space-time tradeoff for answering range queries (extended abstract). In STOC, pages 128-136. ACM, 1982. Google Scholar
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